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Polariton theory of resonant electronic Raman scattering on neutral donor levels in semiconductors
with a direct band gap
Nguyen Ba An, Nguyen van Hieu, Nguyen Toan Thang, Nguyen Ai Viet
To cite this version:
Nguyen Ba An, Nguyen van Hieu, Nguyen Toan Thang, Nguyen Ai Viet. Polariton theory of resonant electronic Raman scattering on neutral donor levels in semiconductors with a direct band gap. Journal de Physique, 1980, 41 (9), pp.1067-1074. �10.1051/jphys:019800041090106700�. �jpa-00208921�
Polariton theory of resonant electronic Raman scattering
on neutral donor levels in semiconductors with a direct band gap Nguyen Ba An, Nguyen Van Hieu, Nguyen Toan Thang and Nguyen Ai Viet
Institute of Physics, National Center for Scientific Research of Vietnam, Nghia Do, Tu Liem, Hanoi, Vietnam (Reçu le 18 janvier 1980, révisé le 21 avril, accepté le 9 mai 1980)
Résumé. 2014 Dans ce travail nous présentons la théorie de la diffusion de Raman sur les électrons liés aux donneurs neutres dans les semiconducteurs avec une bande interdite directe, quand la fréquence du photon incident est en
résonance avec l’exciton. La section efficace de diffusion du polariton avec transition de l’électron de l’état fonda- mental au premier état excité a été calculée.
Abstract. 2014 The theory is presented for the Raman scattering on bound electrons of neutral donors in the semi- conductors with a direct band gap at the incident photon frequency in the resonance with the exciton. The cross-
section of the polariton scattering with the transition of the donor electron from the ground state to the first
excited state is calculated.
Classification Physies Abstracts
71.35 - 71.36 - 78.20B
1. Introduction. - Different electronic Raman
scattering processes in semiconductors were investi-
gated both theoretically and experimentally by many
authors. In the case of the scattering on a gas of mobile carriers (electrons or holes) there may be the
scattering with the single particle excitation or the
scattering on collective excitations in this gas. The
scattering processes of this type were studied widely by Platzman, Tzoar, Wolff et al. [1-13]. The interband scattering processes, in which the initial and final states of electrons or holes belong to different energy
bands, were studied in many works [14-17]. The
initial and final states of the charge carriers may be also their bound states in the Coulomb field of the donor or acceptor ions. In this case we have the scattering on the impurity levels [18-27]. The role of
different extemal electromagnetic fields on the elec-
tronic Raman scattering processes was also studied in details [28-31]. Recently Weisbuch and Ulbrich have investigated experimentally the resonant electronic
Raman scattering on neutral donor levels at the energy of the incoming photon near that of the exci-
ton [32]. The present work is devoted to the theoretical
investigation of the resonant electronic Raman scatter-
ing on neutral donor levels in a two band semiconduc-
tor with the direct band gap and the allowed electrical
dipole transition from the valence to the conduction band. In the case of the material with a very low concentration of donors the electron-hole pair can
exist in their bound states - the excitons.
Due to the virtual transition between the photons
and the excitons there arises in the semiconductor a new kind of elementary excitations - the polaritons
or more precisely the excitonic polaritons discovered by Hopfield [33] and Pekar [34] and studied further in many works [35-42]. As in the case of the resonant
Raman scattering, at the photon energy near that of the exciton the polariton effect becomes very impor-
tant. The polariton theory of the resonant Raman scattering was developed by Mills, Burstein et al. [43]
and also by Bendow and Birman [44].
In the present work we apply the method of Bendow
and Birman [44] to the study of the resonant electronic
Raman scattering on the neutral donor levels at the energy of the incoming or outgoing photon near that
of an exciton. We assume that the semiconductor has
a direct band gap with the allowed electrical dipole
transition and consider only the case of the material with a high purity, when the donor concentration is very small. In this case there will be no screening of the
Coulomb interaction between the electrons and the
holes, the excitons will exist, and therefore the pola-
riton effect must be taken into account. We shall use
the unit system with h = c = 1.
2. Electromagnetic interaction and polariton scatter- ing. - The existence of the polariton and all the
scattering processes considered in this work are the consequences of the electromagnetic interaction in the semiconductor. We begin our study by discussing the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019800041090106700
1068
physical implications of différent elementary electro- magnetic interaction processes.
One type of these electromagnetic interaction processes in the semiconductor is the Coulomb interaction between the charge carriers. Due to this
interaction an electron-hole pair can form their bound
states as a hydrogen-like (positronium-like) system.
We shall show that many scattering processes involv-
ing excitons and charge carriers are also the conse-
quences of the Coulomb interaction between the
charge carriers (free or bound in the impurity) and the
constituents of the excitons.
Another type of the elementary electromagnetic
interaction processes is the recombination of an
electron-hole pair into a photon or its inverse process - the absorption of a photon followed by the crea-
tion of an electron-hole pair. Because of the Coulomb interaction the electron-hole pair can be created in its bound state - the exciton, and we have the photon -
exciton transition. Due to this transition the photons
and the excitons are no longer the eigenstates of the
Hamiltonian of the exciton-photon system : the eigen-
states describe a new kind of quasiparticles in the
semiconductor - the polaritons, which will be denot- ed by n,(k); a = 1, 2, ... Note that besides the exis- tence of the polaritons the second type of the electro- magnetic interaction is also the origin of many inelastic
collision processes involving electrons, excitons, pho-
tons and contributing to the electronic resonant Raman scattering, as we shall see in the sequel.
In studying the mechanism of the collision pro-
cesses involving two identical particles in the initial and/or the final states - two electrons, one in the donor ion and one in the exciton, it is necessary also to
satisfy the requirement of the Fermi statistics : the state vectors must be antisymmetrical under the interchange of two electrons. The physical implica-
tions of the Fermi statistics can be interpreted as the
contributions of some exchange interaction. The role of this exchange interaction will be also taken into account.
In this work for simplicity we shall consider the
photon in a definite polarization state and denote by a(k) the destruction operator for the photon with the
momentum k and the energy w(k). Due to the electro-
magnetic interaction this photon can be transformed
into différent bound states Exv(k); v = 1, 2, 3, ...
of the electron - hole pair - the excitons. Denote by b,(k) the destruction operators for the exciton in the states Exv(k) with the momentum k and the
energy E,(k), and by gv(k) the effective coupling
constants of the transition between the photon and the
excitons Exv(k). After diagonalizing the Hamiltonian of the exciton-photon system we obtain the explicit expression for the destruction operators Cx(k) of the polaritons n,,(k) [41]
Here the index a denotes the branch of the polariton
whose energy Q(I.(k) is determined by the equation
We assume also that there are only one kind of
electrons and one kind of holes, i.e. suppose that the conduction and valence bands are nondegenerate.
In this case the index v will be the set of the quantum numbers {n, 1, m} of the internal (envelope) wave
function Fv(r) of the exciton as a hydrogen-like sys- tem. For the effective coupling constant g,(k) of the exciton-photon transition we have [41, 42]
where (k) is the electrical dipole transition matrix element, eo is the background dielectric constant.
Denote by eN and eN-, the electrons bound in the donor ion,
and consider the scattering process
From eq. (1) it follows that the matrix element of the process (I) is a linear combination of the amplitudes of the following collision processes involving bare particles-bare excitons and bare photons :
namely
where 9N is the energy of the electron bound in the donor ion.
3. Collision amplitudes for bare particles. - In this section we calculate the matrix elements of the pro-
cesses (II), (IIIa), (IIIb) and (IV). Denote by fv(p) and h’(P) the (envelope) internal wave functions of the excitons Exv and Ex, in the p-space, by ON(p) and ON(P) the Fourier transforms of the wave functions of the bound electrons.
Consider first the scattering of the photons on the electrons (process (II)). In the reference [27] two of the
authors have shown that at the value of the photon energy near that of the band gap the virtual transition between
two bands give main contributions to the matrix element, and we can neglect the contribution of the intraband virtual transitions between the donor levels. The scattering process with the virtual interband transitions can be
represented by the Feynman diagrams in figure la and lb. In comparison with the matrix element of the diagram
in figure la that of the diagram in figure lb is negligible. Denote by k and k’ the photon momenta, by Eg the band
gap, by me and mh the effective masses of the electron (in the conduction band) and the hole (in the valence band), by ae the Bohr radius of the donor electron and by aexc that of the exciton. We have
where
Fig. 1. - Feynman diagrams of process (II). Solid line : electron, dashed line : hole, wavy line : photon.
Since the Hamiltonian is hermitic, in the lowest order the matrix elements of the collision processes (IIIa)
and (IIIb) are complex conjugate one with another
One of them equals
where
The Feynman diagrams of the processes (IIIa) and (IIIb) are represented in figure 2a and figure 2b, respectively.
For this scattering mechanism the exchange effect plays an important role as this can be seen from figures 2a
and 2b.
1070
The inelastic scattering of the excitons - process (IV) can take place due to the Coulomb interaction of the
charge carriers : the donor electron with the electron or the hole in the exciton. Taking into account both direct and exchange effect we obtain following result for its matrix element :
u
Fig. 2. - Feynman diagrams of processes (Illa) and (Illb). Double line consisting of a solid line and a dashed one : exciton.
î
where
In the formulaes (11)-(15) = aexclae and the integrals 3’ - Je are defined as follows :
where Ak = k’ - k. The corresponding Feynman diagrams are represented in figures 3a-3e.
From the above results we can calculate the cross-
section of the scattering of the polaritons. At a given
energy there may not exist or there may exist some
polaritons, which belong to different polariton
branches characterized by the index oc. Denote by Q xN the cross-section of the scattering process (I) in
Fig. 3. - Feynman diagrams of process (IV). Double wavy line : Coulomb interaction. a) Direct Coulomb interaction between the donor electron and the electron in the exciton; b) Direct
Coulomb interaction between the donor electron and the hole in the exciton ; c) Coulomb interaction between the donor electron and the electron in the exciton together with the exchange ; d) and e) Coulomb interaction between the electron and the hole together with the exchange.
which the initial and final polaritons belong to the
branches a and a’, we have
where k’ is determined by the equation
and M ’!’ (k, k’) is the following sum
If the probability of the presence of the given polariton
branch a in the incoming polariton beam with the
given energy is denoted by Px, then for the total scattering cross-section of the polariton in the beam to the final state with energy satisfying the conserva-
tion law we have the expression
4. An example. - As an illustration of the general
results obtained above we consider the polariton
effect in the resonant electronic Raman scattering on
neutral donor levels in GaAs taking into account only the contribution of the discrete exciton states
which are the bound states of an electron and a heavy
hole. We use following values of the effective masses
of the electron and the hole :
and take
For the dipole transition matrix element we use the
experimental date given in reference [45]. From
eq. (4) we obtain following values of the effective
coupling constants of the photon-exciton transition :
At the value of the energy of the incoming polariton
near that of the band gap Eg the matrix elements
RNN’, RvNN’ and Rvv’NN, are almost constant and equal :
For the exciton state v = v’ = 1s and for the bound electron states N = ls, N’ = 2s we have following
values
We see that the matrix elements-RNN’, RvNN’ and RB,B"NN’
have the values of the same order. Their contribution to the matrix element of the scattering of the polari-
tons depend also upon the transformation coefficients
ux(k) and v,,,(k).
Consider the special case when only one exciton
state v = 1 s is taken into account, and the dispersion
of the exciton is neglected
In this case there are two polariton branches a = 1, 2 (Fig. 4) with energies
1072
Fig. 4. - Polariton branches.
The transformation coefficients are determined by equations
where
The polariton scattering matrix element becomes
It can be verified that at large values of the polariton
momentum k the coefficients u 1 (k) and V 2(k) are
very small and the corresponding terms in eq. (30)
can be neglected. Now let the initial and final states of the bound electron be the 1 s- and 2s-states. Denote
by A6 the difference of their energies
by l5 = gîslEls s the gap in the polariton energy spec- trum. If the energy of the incoming polariton is
smaller than £ls, then there is only the polariton in
the first (lower) branch oc = 1, and the total cross-
section (24) equals the cross-section aii. For the
incoming polariton energies in the range from El, S + d
to E1s S + l5 + A6 there is only the polariton of the
second (upper) branch a = 2 in the incoming polari-
ton beam, but the final polariton must belong to the
first (lower) branch ; in this case the total cross-section
equals a 21’ Finally, at the incoming polariton energies larger than E1s + l5 + A8 both initial and final
polaritons belong to the second (upper) branch
a = a’ = 2 and we have the cross-section (J22. Due to the presence of the factor
which vanishes at the values of the polariton momen-
tum such that the energy of the incoming polariton equals E1s, E1s S + £5 and E ls S + £5 + LBE, in the deno- minator of the expression (21) of the cross-section,
the latter diverges at these values of the incoming polariton energy. The curve in figure 5 represents the behaviour of the total cross-section (24) as a function
Fig. 5. - Scattering cross-sections with (solid curve) and without (dashed curve) polariton effect.
of the difference between the incoming polariton
energy and that of the dispersionless Is-exciton.
The relative contributions of the scattering mecha-
nisms corresponding to the processes (II), (III) and (IV)
to the polariton scattering matrix element for N = ls and N’ = 2s are represented by the curves in figure 6.
If the dispersion of the exciton is not negligible,
then there exist the energy intervals where two pola-
ritons from two different branches have the same
energy. This case also can be considered, as it was done in reference [44].
Fig. 6. - Relative contributions of different scattering mechan isms to the total matrix element.
5. Discussion. - In this concluding section we
discuss on some aspects related to the above results.
First, note that we have studied the case when
the initial and final polaritons are the mixing of the photon and the same exciton state v = ls. Thete exists, however, an infinite number of exciton states
with the discrete energies, and it may happen that the
initial photon is in the resonance with one exciton state, v = 2s for example, but the final photon is in
the resonance with the other, v’ = 1 s. We can gene- ralize above results to this case. Now the matrix ele- ment (23) becomes
More precisely we can use the general expression (23)
for the matrix element and eqs. (2) and (3) for the
transformation coefficients and the polariton energies,
but in the sums over v and v’ we take only the terms
with v = 2s and v’ = 1 s.
For the integrals IvNN’ and Jvv’NN’ we have the follow- ing values :
Secondly, we have investigated the scattering on
bound electrons in a donor ion. In this case the momentum is not conserved, as it is easily seen from
the expressions of the scattering matrix elements :
eqs. (5), (6), (8) and (10). These matrix elements are nonzero for any momenta k and k’ of the incoming
and outgoing photons, which are related one with
another by the energy conservation law
It is straightforward to modify our results to apply
to the case of the scattering on free electrons. In this
case the (free) electron wave functions in the p-space
are the b-functions
q and q’ being the momenta of the initial and final electrons. Substituting the expressions (34) for the
wave functions into the r.h.s. of eqs. (7), (9), (16)-(20),
we obtain the expressions for the scattering matrix
elements proportional to the ô-function
which means that the total momentum is conserved.
Third, from the behaviour of the curves in figure 6
it follows that in the resonance region
the exciton-photon mixing is very essential, and the polariton effect is important. In particular, at the
energy range
the amplitude of the scattering of the light is determin-
ed completely in terms of the exciton-electron scatter-
ing amplitude. For the light with the frequency w = Q
such that
the mixing between the photon and the exciton is
weak, and the scattering of the bare photons (pro-
cess (II)) is the most important mechanism contribut- ing to the matrix element of the Raman scattering
of the polaritons. In order to evaluate the role of the
polariton effect we compare the scattering cross-
section calculated above for the polaritons with that
for the bare photons (see Fig. 5).
Acknowledgments. - This work was initiated by
the discussions with Drs C. Weisbuch and R. G.
Ulbrich during the stay of one of the authors (N.V.H.)
at the Ecole Polytechnique, Paris. For the hospitality
and the fruitful discussions we express our deep gratitude to Prof. C. Lampel and Drs. C. Weisbuch and R. G. Ulbrich. For the interest in this work we
thank also Profs. J. In. Birman and H. Z. Cummins at the New York City College.
References
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